Well, get the domain of h(x) first. Since x is in the radical sign and we don't want any negative numbers in there for even indices (e.g. 2), the domain of h(x) is x>=0. For g(x), it is actually the set of real numbers, since it is a polynomial. So there isn't any case for g(x).
Therefore, the only condition is for x to be >=0. So the domain of h(g(x)) is x>=0.